A correct model of media with a microstructure (according to Mindlin's definition), which is defined by the presence of free strains and generalizes the well-known Mindlin, Cosserat and Aero-Kuvshinskii models, is proposed. The correctness of the formulation of the model is determined by using a "kinematic" variational principle, based on a systematic formal description of the kinematics of media, formulation of the kinematic constraints for media of various complexity and the construction of the corresponding strain potential energy using a Lagrange multiplier procedure. A system of constitutive relations is established, and a consistent statement of the boundary-value problem is formulated. It is shown that the model of a medium investigated not only reflects scale effects that are similar to cohesive interactions, but also provides a basis for describing a broad spectrum of adhesive interactions. An interpretation of the physical characteristics responsible for non-classical effects is proposed in the context of an analysis of the physical aspects of the model, and a description of the spectrum of adhesion mechanical parameters is given. Β©2009.